The triangle \(\partial L\)  is compact;
there are finitely many sets
\(H_{x_1},\ldots , H_{x_s}\)  which cover
\(\partial L\) .
The union
\(H_{x_1}\cup \ldots \cup H_{x_s}\) 
form “an interior collar”
for \(\partial L\)  in \(L\) .
For each \(i=1,\ldots , s\) ,
there exists a Cantor set
\(K_i\subset H_{x_i}\) 
such that
\(p_{\ell } (K_i ) = p_{\ell } ( H_{x_i})\) 
for each straight line \(\ell \) 
(\(K_i\)  is an affine image of the set
constructed in [1]}).
The union \(K=K_1\cup \ldots \cup K_s\) 
is the desired set.
